Abstract

The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fractional boundary value problem

D0+αu(t)+f(t,u(t))=0,0<t<1,u(0)=u(1)=u′(0)=0,

where 2 < α ≤ 3 and D0+α is the Riemann-Liouville fractional derivative.

Our analysis relies on a fixed-point theorem in partially ordered metric spaces. The autonomous case of this problem was studied in the paper [Zhao et al., Abs. Appl. Anal., to appear], but in Zhao et al. (to appear), the question of uniqueness of the solution is not treated.

We also present some examples where we compare our results with the ones obtained in Zhao et al. (to appear).

2010 Mathematics Subject Classification: 34B15

Keywords

fractional boundary value problemfixed-point theorempositive solution

1 Introduction

Differential equations of fractional order occur more frequently on different research areas and engineering such as physics, chemistry, economics, etc. Indeed, we can find numerous applications in viscoelasticity, electrochemistry control, porous media, electromagnetic, etc. [1–6].

For an extensive collection of results about this type of equations, we refer the reader to the monograph by Kilbas and Trujillo [7], Samko, Kilbas, and Marichev [8], Miller and Ross [9], and Podlubny [10].

Put ψ(x) = x - φ(x). As φ∈J, this means that ψ∈F and from the last inequality

d(Tu,Tv)≤d(u,v)-ψ(d(u,v)).

This proves that T satisfies the contractive condition of Theorem 1.

Finally, the nonnegative character of the function G(t, s) and f(t, x) [assumption (H1)] gives us

(T0)(t)=∫01G(t,s)f(s,0)ds≥0,

where 0 denotes the zero function.

Therefore, Theorem 2 says us that Problem (2) has a unique nonnegative solution.

□

In the sequel, we present a sufficient condition for the existence and uniqueness of positive solutions for Problem (2) (positive solution means x(t) > 0 for t∈ (0, 1)). The proof of this condition is similar to the proof of Theorem 2.3 of [23]. We present this proof for completeness.

Firstly, notice that x(t) is a fixed point of the operator (Tu)(t)=∫01G(t,s)f(s,u(s))ds and, consequently,

x(t)=∫01G(t,s)f(s,x(s))ds.

Now, suppose that there exists 0 < t* < 1 such that x(t*) = 0. This means that

x(t*)=∫01G(t*,s)f(s,x(s))ds=0.

Using that x(t) is a nonnegative function, f(t, y) is nondecreasing with respect to the second argument and the nonnegative character of G(t, s), we get

0=x(t*)=∫01G(t*,s)f(s,x(s))ds≥∫01G(t*,s)f(s,0)ds≥0.

This gives us x(t*)=∫01G(t*,s)f(s,0)ds=0.

As G(t, s) ≥ 0 and f(s, 0) ≥ 0, the last expression implies

G(t*,s)f(s,0)=0a.e(s).

As G(t*, s) ≠ 0 a.e (s) (because G(t*, s) is given by a polynomial), we can obtain

f(s,0)=0a.e(s).

(6)

On the other hand, as f(t0, 0) ≠ 0 for certain t0∈ [0, 1], the nonnegative character of f(t, y) gives us f(t0, 0) > 0. As f(t, y) is a continuous function, we can find a set A⊂ [0, 1] with t0∈A, μ(A) > 0, where μ is the Lebesgue measure and f(t, 0) > 0 for any t∈A. This contradicts (6).

Therefore, x(t) > 0 for t∈ (0, 1). This finishes the proof. □

Remark 3. In Theorem 4, the condition f(t0, 0) ≠ 0 for certain t0∈ [0, 1] seems to be a strong condition in order to obtain a positive solution for Problem (2), but when the solution is unique, we will see that this condition is very adjusted one. In fact, suppose that Problem (2) has a unique nonnegative solution x(t) then

f(t,0)=0 for each t∈[0,1] if and only if x(t)≡0.

In fact, if f(t, 0) = 0 for each t∈ [0, 1], it is easily seen that the zero function satisfies Problem (2) and the uniqueness of the solution gives us x(t) = 0. The reverse implication is obvious.

Remark 4. Notice that the hypotheses in Theorem 3 are invariant by continuous perturbation. More precisely, if f(t, 0) = 0 for any t∈ [0, 1] and f satisfies (H1) and (H2) of Theorem 3 then g(t, x) = a(t) + f(t, x) with a : [0, 1] → [0, ∞) continuous and a ≠ 0, satisfies assumptions of Theorem 4, and this means that the following boundary value problem

D0+αu(t)+g(t,u(t))=0,0<t<1u(0)=u(1)=u′(0)=0

has a unique positive solution.

Now, we present an example that illustrates our results.

Example 1. Consider the boundary value problem

D0+52u(t)+c+λ⋅arctgu(t)=0,0<t<1,c,λ>0u(0)=u(1)=u′(0)=0

(7)

In this case, α=52 and f(t, u) = c + λ · arctg u. It is easily seen that f(t, u) satisfies (H1) of Theorem 3.

In the sequel, we prove that f(t, u) satisfies (H2) of Theorem 3.

Previously, we consider the function ϕ : [0, ∞) → [0, ∞) given by ϕ(u) = arctg u and we will see that ϕ satisfies

4 Some remarks

In a recent paper [18], the authors study the existence of positive solutions of a particular case of Problem (2). More precisely, they study the following fractional autonomous boundary value problem

D0+αu(t)+λf(u(t))=0,0<t<1u(0)=u(1)=u′(0)=0,

(8)

where 2 < α ≤ 3, λ is a positive parameter and f : (0, ∞) → (0, ∞) is continuous. The main tool used by the authors in this paper is Guo-Kranosel'skii fixed-point theorem on cones. In [18], the question about the uniqueness of solutions is not treated.

On the other hand, following a similar reasoning that in Example 1, Theorem 4 gives us the existence of a unique positive solution for Problem (9) when 0<λ≤1Γ(52+1)3532-3552-1≈17.8682.

Our main contribution is the uniqueness of positive solution for Problem (9) when 0 < λ ≤ 17.8682.

Now, we present an example that cannot be studied by the results of [18], and it can be treated by the ones obtained in this paper.

Example 3. Consider the following boundary value problem

D0+5∕2u(t)+λ(t+arctgu(t))=0,0<t<1,λ>0,u(0)=u(1)=u′(0)=0,

(10)

In this case, the boundary value problem is nonautonomous, and thus, this problem cannot be studied by the results of [18].

On the other hand, using a similar argument that in example 1, and using Theorem 4, we obtain the existence of a unique positive solution for Problem (10) when 0 < λ ≤ 17.868.

Declarations

Acknowledgements

This research was partially supported by "Ministerio de Educación y Ciencia" Project MTM 2007/65706.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

We are part of the same research group and work together therefore, we can affirm that the contents of this paper has been prepared by all the authors: JC, JH, and KS. All authors read and approved the final manuscript.

Authors’ Affiliations

(1)

Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria

Copyright

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